Weighted discriminators for GNSS BOC signal tracking

Abstract

Modern Global Navigation Satellite System including Galileo and GPS III will employ multiplexed binary offset carrier (MBOC) modulation to achieve spectrum separation and enhanced tracking performance. A challenge of the MBOC or BOC signal tracking is the presence of ambiguities due to multiple sidepeaks of the autocorrelation functions. Several different techniques including multi-correlator and double estimator schemes have been proposed to address the ambiguity issue. We propose a class of ambiguity-free code tracking techniques by exploiting the unique features of the BOC modulation. In the proposed architecture, the incoming BOC-modulated signals are correlated with BOC-modulated replica and the spreading codes, respectively. Through a multiplicative combination strategy of the two correlator outputs, a noncoherent weighted discriminator is formed and shown to possess the ambiguity-free property. The multipath effect is assessed and compared with existing early-minus-late power and autocorrelation sidepeak cancellation technique discriminators. The noise effects of the theory and simulation are also discussed. In order to further verify the proposed scheme, a set of field data of a Galileo in-orbit validation satellite is collected and processed. It is demonstrated that the proposed method is simple to implement, free from ambiguities, and yields acceptable performance in the presence of multipath and noise.

Appendix: Derivation of the tracking error variance

The appendix derives the tracking error variance of the code tracking loop in the presence of additive Gaussian noise. The derivation is similar to the procedure adopted in (Holmes 1999; Holmes 2007; Julien 2005). Neglecting the multipath term, after substituting (5), (6), (10), and (11) into (24), the weighted discriminator can be expressed as

$$ \begin{aligned} y_{\text{weighted}} (\delta ) = & C\left[ {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right] \\ & + \sqrt C \left[ {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left[ {(v_{{I_{B1} }} - v_{{I_{B2} }} )\cos \phi_{0} + (v_{{Q_{B1} }} - v_{{Q_{B2} }} )\sin \phi_{0} } \right] \\ & + \sqrt C \left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right]\left[ {(w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} )\cos \phi_{0} + (w_{1} v_{{Q_{W1} }} + w_{2} v_{{Q_{W2} }} )\sin \phi_{0} } \right] \\ & + (w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} )(v_{{I_{B1} }} - v_{{I_{B2} }} ) + (w_{1} v_{{Q_{W1} }} + w_{2} v_{{Q_{W2} }} )(v_{{Q_{B1} }} - v_{{Q_{B2} }} ) \\ \end{aligned} $$

(33)

In evaluating the tracking error variance, it can be assumed, without loss of generality, that $\phi_{0} \approx 0$. Thus, the weighted discriminator can be rewritten as

$$ \begin{aligned} y_{\text{weighted}} (\delta ) = & C\left\{ {\left[\vphantom{\frac{d}{2}} {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right]} \right\} \\ & + \left\{ {\left( {w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} } \right)\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) + \left( {w_{1} v_{{Q_{W1} }} + w_{2} v_{{Q_{W2} }} } \right)\left( {v_{{Q_{B1} }} - v_{{Q_{B2} }} } \right)} \right\} \\ & + \sqrt C \left\{ \begin{gathered} \left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right]\left( {w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} } \right) \\ + \left[ {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ \end{gathered} \right\} \\ \end{aligned} $$

(34)

For convenience, the expression (34) is expressed as

$$ \begin{aligned} y_{\text{weighted}} (\delta ) = & C\left\{ {\left[\vphantom{\frac{d}{2}} {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right]} \right\} \\ & + N_{1} + \sqrt C N_{2} \\ \end{aligned} $$

(35)

where

$$ N_{1} = \left( {w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} } \right)\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) + \left( {w_{1} v_{{Q_{W1} }} + w_{2} v_{{Q_{W2} }} } \right)\left( {v_{{Q_{B1} }} - v_{{Q_{B2} }} } \right) $$

(36)

$$ \begin{aligned} N_{2} = \,& \left( {w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} } \right)\left[ {\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)} \right] \\ & + \left[ {w_{1} \Uppsi (\delta - \alpha_{1} T_{c} ) + w_{2} \Uppsi (\delta - \alpha_{2} T_{c} )} \right]\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ \end{aligned} $$

(37)

From (Julien 2005), the tracking error variance is given by

$$ \sigma_{\text{weighted}}^{2} =\, \frac{{2B_{L} \left( {1 - 0.5B_{L} T} \right)T\sigma_{{{\text{y}}_{\text{weighted}} }}^{2} }}{{K_{{{\text{y}}_{\text{weighted}} }}^{2} }} $$

(38)

where $K_{{y_{\text{weighted}} }}$ is the loop gain associated with the weighted discriminator, and $\sigma_{{{\text{y}}_{\text{weighted}} }}$ is the discriminator output standard deviation. The loop gain can be computed as

$$ \begin{aligned} K_{{y_{\text{weighted}} }} = & \left. {\frac{{dy_{\text{weighted}} \left( \delta \right)}}{d\delta }} \right|_{\delta = 0} \\ = & C\left( {w_{1} \Uppsi^{'} ( - \alpha_{1} T_{c} ) + w_{2} \Uppsi^{'} ( - \alpha_{2} T_{c} )} \right)\left( {\Uplambda \left( { - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\frac{d}{2}T_{c} } \right)} \right) \\ & + C\left( {w_{1} \Uppsi ( - \alpha_{1} T_{c} ) + w_{2} \Uppsi ( - \alpha_{2} T_{c} )} \right)\left( {\Uplambda^{'} \left( { - \frac{d}{2}T_{c} } \right) - \Uplambda^{'} \left( {\frac{d}{2}T_{c} } \right)} \right) \\ \end{aligned} $$

(39)

Note that the correlation function of $\Uplambda ( \cdot )$ is symmetric, that is,

$$ \Uplambda \left( { - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\frac{d}{2}T_{c} } \right) = 0 $$

(40)

Thus, the gain can be simplified as

$$ K_{{y_{\text{weighted}} }} = C\Upgamma (0)\left( {\Uplambda^{'} \left( { - \frac{d}{2}T_{c} } \right) - \Uplambda^{'} \left( {\frac{d}{2}T_{c} } \right)} \right) $$

(41)

where $\Uplambda^{'} \left( { - \frac{d}{2}T_{c} } \right) = \left. {\frac{{{\text{d}}\Uplambda \left( {\delta - \frac{d}{2}T_{c} } \right)}}{{{\text{d}}\delta }}} \right|_{\delta = 0}$ and $\Uplambda^{'} \left( {\frac{d}{2}T_{c} } \right) = \left. {\frac{{{\text{d}}\Uplambda \left( {\delta + \frac{d}{2}T_{c} } \right)}}{{{\text{d}}\delta }}} \right|_{\delta = 0}$. Substituting (8) into (41) and after some manipulations, the slope can be shown to be equal to

$$ K_{{y_{\text{weighted}} }} = 4\pi C\Upgamma (0)\int\limits_{ - \infty }^{\infty } {fH(f)S_{g} (f)\sin \left( {\pi fdT_{c} } \right){\kern 1pt} {\text{d}}f} $$

(42)

From (35), the variance of weighted discriminator can be expressed as

$$ \sigma_{{{\text{y}}_{\text{weighted}} }}^{2} = \sigma_{{N_{1} }}^{2} + C\sigma_{{N_{2} }}^{2} + 2\sqrt C \sigma_{{N_{1} N_{2} }} $$

(43)

The variance of $N_{1}$ is given by

$$ \sigma_{{N_{1} }}^{2} = E\left\{ {N_{1} \left( t \right)N_{1} \left( {t - x} \right)} \right\} - E\left\{ {N_{1} \left( t \right)} \right\}E\left\{ {N_{1} \left( {t - x} \right)} \right\} $$

(44)

Let $A_{1} = w_{1} v_{{I_{W1} }} \left( t \right) + w_{2} v_{{I_{W2} }} \left( t \right)$ and $A_{2} = v_{{I_{B1} }} \left( t \right) - v_{{I_{B2} }} \left( t \right)$. Because the Q components are identically distributed and independent from the I components, the terms in (44) can be calculated as

$$ E\left\{ {N_{1} \left( t \right)N_{1} \left( {t - x} \right)} \right\} = 2E\left\{ {A_{1}^{2} } \right\}E\left\{ {A_{2}^{2} } \right\} + 6\left( {E\left\{ {A_{1} A_{2} } \right\}} \right)^{2} $$

(45)

$$ E\left\{ {N_{1} \left( t \right)} \right\}E\left\{ {N_{1} \left( {t - x} \right)} \right\} = 4\left( {E\left\{ {A_{1} A_{2} } \right\}} \right)^{2} $$

(46)

Assuming that the code delay error is small, $N_{2}$ can be reduced to

$$ \begin{aligned} N_{2} = & \left( {w_{1} v_{{I_{W1} }} + w_{2} v_{{I_{W2} }} } \right)\left[ {\Uplambda \left( { - \frac{d}{2}T_{c} } \right) - \Uplambda \left( {\frac{d}{2}T_{c} } \right)} \right] + \left[ {w_{1} \Uppsi ( - \alpha_{1} T_{c} ) + w_{2} \Uppsi ( - \alpha_{2} T_{c} )} \right]\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ = & \left[ {w_{1} \Uppsi ( - \alpha_{1} T_{c} ) + w_{2} \Uppsi ( - \alpha_{2} T_{c} )} \right]\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ \end{aligned} $$

(47)

Owing to (40), $N_{2}$ can be simplified as:

$$ \begin{aligned} N_{2} = \,& \left[ {w_{1} \Uppsi ( - \alpha_{1} T_{c} ) + w_{2} \Uppsi ( - \alpha_{2} T_{c} )} \right]\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ =\, & \Upgamma \left( 0 \right)\left( {v_{{I_{B1} }} - v_{{I_{B2} }} } \right) \\ \end{aligned} $$

(48)

In a similar way, the variance of $N_{2}$ can be expressed as:

$$ \sigma_{{N_{2} }}^{2} = \Upgamma^{2} \left( 0 \right)E\left\{ {A_{2}^{2} } \right\} $$

(49)

The results of $E\left\{ {A_{1} A_{2} } \right\}$, $E\left\{ {A_{1}^{2} } \right\}$, and $E\left\{ {A_{2}^{2} } \right\}$ can be derived by using the spectral characteristic of the input signal/noise. The covariance at each correlator can be expressed as:

$$ E\left\{ {v_{{I_{Bk} }} \left( t \right)v_{{I_{Bl} }} \left( {t - x} \right)} \right\} = \frac{{N_{0} }}{2T}\tilde{R}_{g} \left( {\left( {d_{l} - d_{k} } \right)T_{c} } \right) $$

(50)

$$ E\left\{ {v_{{I_{Bk} }} \left( t \right)v_{{I_{Wl} }} \left( {t - x} \right)} \right\} = \frac{{N_{0} }}{2T}\tilde{R}_{x} \left( {\left( {\alpha_{l} - d_{k} } \right)T_{c} } \right) $$

(51)

$$ E\left\{ {v_{{I_{Wk} }} \left( t \right)v_{{I_{Wl} }} \left( {t - x} \right)} \right\} = \frac{{N_{0} }}{2T}\tilde{R}_{c} \left( {\left( {\alpha_{l} - \alpha_{k} } \right)T_{c} } \right) $$

(52)

where

$$ \tilde{R}_{g} \left( \tau \right) = \int\limits_{ - \infty }^{\infty } {S_{g} (f)\left| {H(f)} \right|^{2} {\text{e}}^{j2\pi f\tau } {\kern 1pt} {\text{d}}f} $$

(53)

$$ \tilde{R}_{x} \left( \tau \right) = \int\limits_{ - \infty }^{\infty } {S_{x} (f)\left| {H(f)} \right|^{2} {\text{e}}^{j2\pi f\tau } {\kern 1pt} {\text{d}}f} $$

(54)

$$ \tilde{R}_{c} \left( \tau \right) = \int\limits_{ - \infty }^{\infty } {S_{c} (f)\left| {H(f)} \right|^{2} {\text{e}}^{j2\pi f\tau } {\kern 1pt} {\text{d}}f} $$

(55)

In the above, $S_{c} ( \cdot )$ is the power spectral density of the PRN waveform which is given by $S_{c} (f) = T_{c} {\text{sinc}}^{2} \left( {\pi fT_{c} } \right)$.

The covariance between $N_{1}$ and $N_{2}$ is zero since the product of $N_{1}$ and $N_{2}$ leads to a polynomial with monomials of odd order in zero mean Gaussian random variables. Then,

$$ \sigma_{{N_{1} N_{2} }} = 0 $$

(56)

After some calculations, (44) and (49) can be, respectively, written as:

$$ \sigma_{{N_{1} }}^{2} = 2\left( {\frac{{N_{0} }}{2T}} \right)^{2} \left( \begin{gathered} 2\left( {\tilde{R}_{g} \left( 0 \right) - \tilde{R}_{g} \left( {dT_{c} } \right)} \right) \times \left( {w_{1}^{2} \tilde{R}_{c} \left( 0 \right) + 2w_{1} w_{2} \tilde{R}_{c} \left( {\left( {\alpha_{2} - \alpha_{1} } \right)T_{c} } \right) + w_{2}^{2} \tilde{R}_{c} \left( 0 \right)} \right) \\ + \left[ \begin{gathered} w_{1} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{1} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{1} } \right)T_{c} } \right)} \right) \hfill \\ + w_{2} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{2} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{2} } \right)T_{c} } \right)} \right) \hfill \\ \end{gathered} \right]^{2} \\ \end{gathered} \right) $$

(57)

and

$$ \begin{aligned} \sigma_{{N_{2} }}^{2} = & \frac{{N_{0} }}{T}\Upgamma \left( 0 \right)^{2} \left( {\tilde{R}_{g} \left( 0 \right) - \tilde{R}_{g} \left( {dT_{c} } \right)} \right) \\ = & 2\frac{{N_{0} }}{T}\Upgamma \left( 0 \right)^{2} \int\limits_{ - \infty }^{\infty } {S_{g} (f)\left| {H(f)} \right|^{2} \sin^{2} \left( {\pi fdT_{c} } \right){\kern 1pt} {\text{d}}f} \\ \end{aligned} $$

(58)

Using (56), (57), and (58), the variance of the weighted discriminator output can be expressed as:

$$ \begin{aligned} \sigma_{{y_{\text{weighted}} }}^{2} =\, & 2CN_{0} \Upgamma \left( 0 \right)^{2} \frac{1}{T}\int\limits_{ - \infty }^{\infty } {S_{g} (f)\left| {H(f)} \right|^{2} \sin^{2} \left( {\pi fdT_{c} } \right){\kern 1pt} {\text{d}}f} \\ & \times \left( \begin{gathered} 1 + \frac{{\left( {w_{1}^{2} \tilde{R}_{c} \left( 0 \right) + 2w_{1} w_{2} \tilde{R}_{c} \left( {\left( {\alpha_{2} - \alpha_{1} } \right)T_{c} } \right) + w_{2}^{2} \tilde{R}_{c} \left( 0 \right)} \right)}}{{\left( {{C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-\nulldelimiterspace} {N_{0} }}} \right)T\Upgamma \left( 0 \right)^{2} }} \hfill \\ + \frac{{\left[ \begin{gathered} w_{1} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{1} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{1} } \right)T_{c} } \right)} \right) \hfill \\ + w_{2} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{2} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{2} } \right)T_{c} } \right)} \right) \hfill \\ \end{gathered} \right]^{2} }}{{2\left( {{C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-\nulldelimiterspace} {N_{0} }}} \right)T\Upgamma \left( 0 \right)^{2} \left( {\tilde{R}_{g} \left( 0 \right) - \tilde{R}_{g} \left( {dT_{c} } \right)} \right)}} \hfill \\ \end{gathered} \right) \\ \end{aligned} $$

(59)

Assuming a front-end filter with unity gain within $\pm {{B_{w} } \mathord{\left/ {\vphantom {{B_{w} } 2}} \right. \kern-\nulldelimiterspace} 2}$ Hz, and substituting (42) and (59) into (38), the code tracking error variance is obtained.

$$ \sigma_{\text{weighted}}^{2} = \frac{{B_{L} \left( {1 - 0.5B_{L} T} \right) \times \int_{{ - {{B_{w} } \mathord{\left/ {\vphantom {{B_{w} } 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{{B_{w} } \mathord{\left/ {\vphantom {{B_{w} } 2}} \right. \kern-\nulldelimiterspace} 2}}} {S_{g} (f)\left| {H(f)} \right|^{2} \sin^{2} \left( {\pi fdT_{c} } \right){\kern 1pt} {\text{d}}f} }}{{\left( {{C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-\nulldelimiterspace} {N_{0} }}} \right)\left( {2\pi \int_{{ - {{B_{w} } \mathord{\left/ {\vphantom {{B_{w} } 2}} \right. \kern-\nulldelimiterspace} 2}}}^{{{{B_{w} } \mathord{\left/ {\vphantom {{B_{w} } 2}} \right. \kern-\nulldelimiterspace} 2}}} {fH(f)S_{g} (f)\sin \left( {\pi fdT_{c} } \right){\kern 1pt} {\text{d}}f} } \right)^{2} }} \times \left( {1 + \frac{{\mu_{1} + \mu_{2} }}{{\left( {{C \mathord{\left/ {\vphantom {C {N_{0} }}} \right. \kern-\nulldelimiterspace} {N_{0} }}} \right)T}}} \right) $$

(60)

where

$$ \mu_{1} = \frac{{\left( {w_{1}^{2} \tilde{R}_{c} \left( 0 \right) + 2w_{1} w_{2} \tilde{R}_{c} \left( {\left( {\alpha_{2} - \alpha_{1} } \right)T_{c} } \right) + w_{2}^{2} \tilde{R}_{c} \left( 0 \right)} \right)}}{{\Upgamma \left( 0 \right)^{2} }} $$

(61)

$$ \mu_{2} = \frac{{\left[ {w_{1} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{1} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{1} } \right)T_{c} } \right)} \right) + w_{2} \left( {\tilde{R}_{x} \left( {\left( {\frac{d}{2} - \alpha_{2} } \right)T_{c} } \right) - \tilde{R}_{x} \left( {\left( { - \frac{d}{2} - \alpha_{2} } \right)T_{c} } \right)} \right)} \right]^{2} }}{{2\left( {\tilde{R}_{g} \left( 0 \right) - \tilde{R}_{g} \left( {dT_{c} } \right)} \right)\Upgamma \left( 0 \right)^{2} }} $$

(62)